Statement
If x1, x2,..., xd are d complex numbers that are linearly independent over the rational numbers, and y1, y2,...,yl are l complex numbers that are also linearly independent over the rational numbers, and if dl > d + l, then at least one of the following dl numbers is transcendental:
The most interesting case is when d = 3 and l = 2, in which case there are six exponentials, hence the name of the result. The theorem is weaker than the related but thus far unproved four exponentials conjecture, whereby the strict inequality dl > d + l is replaced with dl ≥ d + l, thus allowing d = l = 2.
The theorem can be stated in terms of logarithms by introducing the set L of logarithms of algebraic numbers:
The theorem then says that if λij are elements of L for i = 1, 2 and j = 1, 2, 3, such that λ11, λ12, and λ13 are linearly independent over the rational numbers, and λ11 and λ21 are also linearly independent over the rational numbers, then the matrix
has rank 2.
Read more about this topic: Six Exponentials Theorem
Famous quotes containing the word statement:
“No statement about God is simply, literally true. God is far more than can be measured, described, defined in ordinary language, or pinned down to any particular happening.”
—David Jenkins (b. 1925)
“The force of truth that a statement imparts, then, its prominence among the hordes of recorded observations that I may optionally apply to my own life, depends, in addition to the sense that it is argumentatively defensible, on the sense that someone like me, and someone I like, whose voice is audible and who is at least notionally in the same room with me, does or can possibly hold it to be compellingly true.”
—Nicholson Baker (b. 1957)
“The new statement will comprise the skepticisms, as well as the faiths of society, and out of unbeliefs a creed shall be formed. For, skepticisms are not gratuitous or lawless, but are limitations of the affirmative statement, and the new philosophy must take them in, and make affirmations outside of them, just as much as must include the oldest beliefs.”
—Ralph Waldo Emerson (1803–1882)